What exactly do you mean? The alephs are not integers. That $\aleph_1$ comes directly after $\aleph_0$ is by definition. For a natural number $n>0$, $\aleph_n$ is the first aleph that comes after $\aleph_{n-1}$.
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Michael Greinecker♦Sep 26 '12 at 9:56

3 Answers
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The cardinals are the following ones:
$$0,1,2,3,4,5,6,\dots,\aleph_0,\aleph_1,\aleph_2, \aleph_3,\dots,\aleph_\omega,\aleph_{\omega+1},\dots,\aleph_{2\omega},\aleph_{2\omega+1},\dots $$
Where $\aleph_0$ is the first infinite cardinal (the cardinality of each infinite countable set), so $\aleph_0\notin\mathbb N$, is not said to be an integer in the ordinary way. Then $\aleph_1$ is the next cardinal, and so on..
(and this "and so on.." also includes some knowledge about the ordinals).

By Cantor's theorem ($|P(A)| > |A|$ for all sets $A$) we have that for every cardinal there is a bigger cardinal. By the well foundedness and the axiom of choice in ZFC, we have that every cardinal is a cardinal of a well-ordered set (which is in bijection to some ordinal), and it follows that there is always a next cardinal..

The cardinal number $\aleph_1$ is defined as the cardinality of the set of all countable ordinals, i.e. ordinals of cardinality $\le\aleph_0$. If you believe that that set of countable ordinals exists, then you've got it.

Perhaps you are asking for an example of a set of cardinality $\aleph_1$. Cantor thought the set of all real numbers would be an example. In fact, he thought he had a proof. Then he found a hole in the proof, and restated the question as a hypothesis, which has come down to us as "The Continuum Hypothesis": the cardinality of the reals is $\aleph_1$. No one has been able to prove this, nor to disprove it: In 1940, Gödel published his proof that this hypothesis cannot be disproved on the basis of the axioms of mathematics generally accepted at the time and, in 1963, Cohen published his proof that it cannot be proved either.

The bottom line is, no one has ever been able to present a set provably of cardinality $\aleph_1$ and, on our current understanding of these things, no one ever will.

If that doesn't answer your question, perhaps you'd like to edit your question so someone will be able to get what exactly you're asking.

I do not understand what you mean, unless you simply mean a set of reals. Because one can exhibit sets of size $\aleph_1$. There are reasonable equivalence relations on the reals that have precisely $\aleph_1$ classes, for example.
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Andres CaicedoSep 27 '12 at 6:59

I think for historical accuracy, one should change the sentence on Cohen. Cohen showed that CH cannot be proven. That CH cannot be disproven has been shown before by Gödel.
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Michael Greinecker♦Sep 27 '12 at 9:42

@Andres, I did have sets of reals in mind, but I confess to being unaware of the kind of example you are talking about. Perhaps you could post an answer giving a set of cardinality $\aleph_1$, as that might be what OP is looking for (and as I might learn something from it).
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Gerry MyersonSep 27 '12 at 10:36

@Michael, I opted for brevity. If it bothers you, feel free to edit.
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Gerry MyersonSep 27 '12 at 10:38

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(Gerry, I made the edit.) Anyway, a simple example is the following: We can easily identify a real with a sequence of naturals. For example, an irrational between $0$ and $1$ has an infinite continued fraction. Each natural can be seen as coding two. For example, $2^a(2b+1)$ could code $(a,b)$. This way, each real can be seen as coding a binary relation. Identify two reals iff they do not code well-orderings, or else, they code well-orderings of the same order type. This relation has precisely $\omega_1$ equivalence classes. (On the other hand, finding a "coding free" relation seems harder.)
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Andres CaicedoSep 27 '12 at 22:59